Generalizing Path-Integration

The path-integral formulation of quantum mechanics determines particle propagators through a certain functional integral, averaging the motion of a particle over all possible paths. Such an integral is defined, by the time-discretization model, by slicing the total time-interval into an integer number of "time segments", calculating a multiple integral for each such discretization, and taking a limit as the slicing index approaches infinity. Thus, the quantum path-integral is fundamentally defined as a limit of a positive integer index.

The Multiphase Discretization Metaverse Model, or QCI, generalizes this procedure to complex values of the index. Imaginary or complex time-intervals have already been considered, but the slicing index, determining the path skeletonization structure, has always been taken to be a real number. When this index itself is complex, the limit defining the path-integral is indeterminate, with a spectrum of distinct propagators corresponding to different limits of the index in the complex plane. These distinct propagators are postulated to apply to different universes.

Although this generalization might seem arbitrary, similar such generalizations have found utility in fundamental physics. For example, Steven Weinberg considered a generalization of the Einstein field equations by introducing a non-zero cosmological constant, with each different value of the constant associated with a distinct universe. Superstring theory considers distinct Calabi-Yau manifolds potentially corresponding to distinct universes. History has shown us that the generalization of "number", from the naturals to the reals to the complex, poses applications in physics. It seems only natural to allow Feynman's limiting index to be complex, and to consider the variety of different results obtained.